What Is the Mode in Statistics?
The mode is the value that appears most often in a data set. To find it, you simply count how many times each value occurs and pick the one with the highest frequency. For example, in the set {2, 5, 2, 6, 7, 3, 2}, the value 2 appears three times — more than any other — so the mode is 2.
The mode is one of the three measures of central tendency, alongside the mean and the median. Unlike the mean (the average) and the median (the middle value), the mode is the only one of the three that can be used with non-numeric, categorical data — such as the most common eye colour or the best-selling shoe size. A data set is not limited to a single mode: it can have one mode, two modes (bimodal), several modes (multimodal), or no mode at all.
“The mode is the most frequently occurring value in a data set. A set of data may have one mode, more than one mode, or no mode at all.” — Australian Bureau of Statistics, Statistical Language Glossary
This guide explains what the mode is, how to calculate it for both ungrouped and grouped data, how to handle data sets with two modes or no mode, and where the mode is most useful in real research.
How Is the Mode Calculated?
The method for finding the mode depends on the kind of data you have. Data can be either ungrouped (a plain list of values) or grouped (values organised into classes with frequencies).
Calculating the Mode for Ungrouped Data
Ungrouped data is just a raw list of values with no particular order. Finding the mode is straightforward — you only need to identify the value that occurs most often:
- List every distinct value in the data set.
- Count how many times each value appears (its frequency).
- The value with the highest frequency is the mode.
For example, the set {2, 5, 2, 6, 7, 3, 2} has no fixed pattern, yet the mode is clearly 2 because it appears three times — more often than any other value. Sorting the list first (2, 2, 2, 3, 5, 6, 7) makes repeated values easier to spot, but it is not required.
Calculating the Mode for Grouped Data
Grouped data is a little trickier. Here the values are arranged in two columns (or rows): one lists the observed values or class intervals, and the other lists the frequencies that correspond to them. For example, the table below shows a grouped data set.
| Class interval (marks) | No. of students (frequency) |
|---|---|
| 2 – 4 | 3 |
| 4 – 6 | 4 |
| 6 – 8 | 2 |
| 8 – 10 | 1 |
For grouped data with class intervals, you cannot read the mode off directly, because individual values are hidden inside the classes. Instead, you first find the modal class (the class with the highest frequency) and then estimate the mode within it using the formula:
Mode = L + [ (f1 − f0) ÷ (2f1 − f0 − f2) ] × h
Where:
- L = the lower limit (lowest value) of the modal class
- h = the size (width) of the class interval
- f1 = the frequency of the modal class
- f0 = the frequency of the class before the modal class
- f2 = the frequency of the class after the modal class
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Some Important Terms for the Mode of Grouped Data
1. Modal class: In grouped data, the class (or interval) that contains the highest frequency — i.e. where a value occurs the greatest number of times — is called the modal class.
2. Class interval: Grouped data sets come in two forms. In one form, the data is grouped into intervals, such as 2–4 or 4–6 in the table above. In the other form there are no intervals — the data is simply grouped into two columns, one holding the observed variables and the other holding the frequencies that correspond to them. For instance, if 10 children visit a park 4 times and 5 children visit it 2 times, the number of visits is the observed value and the number of children is its frequency.
3. Lower limit of the modal class (L): Because a class interval has two boundaries — where it begins and where it ends — the starting (lower) value is called the lower limit of that class interval.
Examples to Calculate the Mode for Grouped Data
Let us work through the grouped data with class intervals shown earlier. The modal class is 4–6, because it has the highest frequency (4). We then read off the values needed for the formula:
- L = 4 (lower limit of the modal class)
- h = 2 (width of each interval)
- f1 = 4 (frequency of the modal class)
- f0 = 3 (frequency of the class before it, 2–4)
- f2 = 2 (frequency of the class after it, 6–8)
Mode = L + [ (f1 − f0) ÷ (2f1 − f0 − f2) ] × h
Mode = 4 + [ (4 − 3) ÷ (2×4 − 3 − 2) ] × 2
Mode = 4 + [ 1 ÷ (8 − 5) ] × 2
Mode = 4 + (1 ÷ 3) × 2
Mode = 4 + 0.667 = 4.67 (to 2 decimal places).
When grouped data has no intervals — just discrete values and their frequencies — you do not need the formula at all. You simply pick the value with the highest frequency, exactly as with ungrouped data. In the table below, the value 13 is the mode because it has the highest frequency (10).
| No. of children taking the bus to school | Frequency |
|---|---|
| 24 | 4 |
| 13 | 10 |
| 2 | 9 |
Bimodal, Multimodal and No-Mode Data Sets
A common exam question is whether a data set can have more than one mode. The answer is yes — the claim that “a data set will always have exactly one mode” is false. Depending on the frequencies, a data set can be:
| Type | Number of modes | Example | Mode(s) |
|---|---|---|---|
| Unimodal | One | {2, 4, 4, 4, 7, 9} | 4 |
| Bimodal | Two | {1, 1, 3, 5, 5, 8} | 1 and 5 |
| Multimodal | Three or more | {2, 2, 4, 4, 6, 6, 9} | 2, 4 and 6 |
| No mode | None | {3, 5, 7, 9, 11} | No mode |
When two values share the highest frequency, the data set is bimodal and has two modes. When three or more values tie for the highest frequency, it is multimodal. And when every value occurs the same number of times — for instance, each value appears exactly once — there is no mode, because no value stands out as the most frequent.
Identifying the mode
Where Is the Mode Used?
The mode is used alongside the mean and median across statistics, mathematics, finance, accounting, academic research, engineering and software design. Its main strengths are:
- Categorical (nominal) data: The mode is the only measure of central tendency that works with nominal data such as gender, colour or brand. You cannot find a meaningful “average” eye colour, but you can find the most common one.
- Spotting the most popular value: Retailers use the mode to find the best-selling product size; transport planners use it to find the most common journey time.
- Resistance to extreme values: Because it depends only on frequency, the mode is unaffected by outliers, which can distort the mean.
- Open-ended classes: The mode can be found even when a frequency table has open-ended intervals (e.g. “56 kg or less”), where the mean cannot be calculated.
In research, the mode is therefore best for describing the typical category of a population or sample. To see how it sits beside the other averages, read our guide to the mean, median and mode, and to understand which data types each average suits, see types of variables.
When Is the Mode Not Calculated?
The mode is less useful — and is often not reported — in two situations:
- When the data is fairly evenly distributed and every value occurs roughly the same number of times. Here there is either no mode or several competing modes, so the mode tells you little about the centre of the data.
- When the data is continuous numeric data for which the mean or median gives a more precise summary. For such data, statisticians usually prefer the mean (for symmetric data) or the median (for skewed data) over the mode.
In these cases the mode is best treated as a supporting statistic rather than the main measure of central tendency.
